 let's talk a little bit more about what it really means factoring polynomials graphically right we already talked about what it means to factor polynomials which is basically just breaking them down to its core elements right but let's let's take this a little bit further and just visually graphically show what it means when you actually take a polynomial and you factor it now when you're factoring numbers just straight out numbers you know we already talked about this in series one right you just take a number whatever integer it is or whatever rational number it is and break it down those prime factors right for example six breaks down to two times three for example the number six you can break down into two times three right and what we end up doing with polynomials is taking a polynomial and breaking it down to its prime polynomials right breaking down to its smaller degree polynomials right for example if we had the following polynomial and we've already talked about this we can break this down in the following form right so if we take the polynomial x squared plus 5x plus 6 we can break it down into x plus 2 times x plus 3 right so all this is really is just one function or original function that we're given if they say factor this polynomial we take this polynomial and break it into two smaller degree polynomials when multiplied together give you the original polynomial right which is really one function is equal to two other functions multiplied together so the way we can think about it is this this polynomial this function we could just call f of x so f of x would be this guy f of x is equal to h of x times g of x and this guy would be our h of x and this guy would be our g of x now this is just basically terminology wise right graphically we can you know present this graphically and graphically this is this is the way it's gonna look like this guy is a parabola so we're taking our parabola and breaking it down into two functions that aren't plot parabolas multiplied together to give the parabola and these two guys here are linear functions their lines so two lines multiplied together give you a parabola and this is what it's gonna look like so our polynomial here x squared plus 5x plus 6 is a parabola and its x intercepts are x is equal to negative 2 and x is equal to negative 3 right we already talked about this when you factor this guy if this guy is equal to 0 all you're doing is finding the x intercepts so you can set each one of these equal to 0 and solve for it right so this is x equals negative 2 and x equals negative 3 so it's a parabola that crosses the x axis as x equals negative 2 and x equals negative 3 so we've got our parabola this guy graphs something like this and we're gonna talk about graphing polynomials you know more more accurately when we get into graphing polynomials when we start talking about completing the square but this guy would be our f of x right this parabola if you break it down into smaller polynomials is going to be two lines multiplied together to give you the original function so what we have here is our h of x is basically y is equal to x plus 2 and again we'll talk about you know graphing linear functions if you look into the polynomial section you know if this is if you're looking at this right now when I'm putting it up haven't done it yet but we will talk about graphing linear functions so this guy x plus 2 is just a line it's a linear function with x y intercept of 2 and a slope of 1 so all you do is you find your y intercept and go up one over one so that graphs a line right the other guy is also linear function a line linear line right with y intercept of 2 and a slope of 1 which is this guy so if you take these two functions multiplying together you get this function if you take these two lines multiplying together you get this parabola here and this should be like hey wait a second how can we multiply two lines and get a curve well if you remember from series one what's two negatives multiply together right they give you a positive so over here we go down into the y equals negative down here you got y is equal to negative right so two negatives multiply together to give you a positive so it kicks the function back up over here between negative 2 and negative 3 here we have part of the function here is negative or this this function here is negative when it goes below negative 2 right when you go past negative 2 on the x intercept in the y section your negative here over here negative 2 is here and that's going to be positive for this function right so negative times a positive over here is going to kick you still in negative section right but as soon as you go past negative 3 or negative 3 here right your y your x intercept is negative 3 as well as soon as you go past your negative 3 here then this function is negative and this function is still negative right so negative and negative kicks you back into the positive in this section right so two lines multiplied together give you a parabola as long as the lines you know they go into both of them go into the negative as long as their their domains whatever their x can be continues on forever right so factoring integers factoring rational numbers is just like this it's just you know two different numbers multiplied together to give you the original number factoring polynomials it's just two other polynomials multiplied together to give you the other polynomial the polynomial you started with graphically it's whatever these polynomials graph now they don't have to be linear functions it could be curves it could be other polynomials right these two polynomials multiplied together give you the original polynomial and that's what factoring is we're taking things and breaking them up and seeing you know what they're made out of and that way you can take you know maybe we want to take this part whatever this thing is and multiply it with another function right this part of the function and multiply by another function just to create a new function right and and you know that's what it is that's what that's what factoring polynomials means you're taking graphically anyway you're taking a function and breaking it up into two other functions multiply together they may be lines they may be parabolas they may be quartic functions cubic functions it could be anything right and they don't even have to be polynomial functions it could be non-polynomial functions right things that have asymptotes things that have you know unknowns vertical or horizontal asymptotes right the things that have breaks that have holes in them right you could take any type of fun any any two functions multiply together to give you a new function and from that new function you can do new things or apply them in new places okay that's what factoring is factoring a polynomial break it down to its function factor rational number break it down to its prime factors right factor polynomial a polynomial is just a graph right it's just a graph of function break it down to its new functions